4 Answers
4

The spell effect would be unchanged.

Any spell can be cast using a higher level spell slot than necessary - so you could cast Find Familiar using a ninth level slot. But the Find Familiar spell doesn't tell you that there would be any additional affects from casting at higher levels - so there wouldn't be any.

Contrast that with spells like Magic Missile (as you suggested) which states explicitly:

At Higher Levels.
When you cast this spell using a spell slot of 2nd level or higher, the spell creates one more dart for each slot level above 1st. (SRD p. 161)

If a spell doesn't have this 'At Higher Levels' section then there are no change to the spell effects from casting it at a higher level.

There are, however, a couple of incidental benefits from casting a spell at higher level:

Your spell will be harder to counterspell than normal (if cast at level 4 or above).

You are still able to cast the spell even if you've run out of low level spell slots.

P.S. The above is the RAW answer to this question and it's what I'd tell my players if they asked.

That said, I have been left pondering if adding some homebrew effects for an NPC villain might be rather fun? You could rule that by casting Find Familiar at ninth level your villain received nine separate standard familairs, or one familiar nine times as strong as usual. Either option could be an interesting character facet for an antagonist.

Likely nothing significant

DMG page 201 states

Casting a Spell at a Higher Level

When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic missile is 2nd level. Effectively, the spell expands to fill the slot it is put into.

Some spells, such as magic missile and cure wounds, have more powerful effects when cast at a higher level, as detailed in a spell's description.

So, unless the spell's description says it does anything at higher levels, as magic missile does, then there are no additional effects of the spell. Since find familiar lacks such a section, there is no resulting difference.

However, the spell being cast "assumes the higher level". So, if another caster happens on someone casting a 9th level find familiar and decides to cast counterspell to cancel it, they're countering a 9th level spell.

Spells that have different or more powerful effects if cast in higher spell slots clearly call out those differences. Find Familiar has no such information so there is no change, regardless of the spell slot.